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A note on speed of convergence to the quasi-stationary distribution

Polak M., Rolski T.

Demonstratio Mathematica

2012

Vol. 45, nr 2

385-397

In this note we show that for Z being a birth and death process on Z or Brownian motion with drift and (…), the speed of convergence to the quasi-stationary distribution is of order 1/t. The corresponding version that X is the number of calls in M/M/1 queue or the reflected Brownian motion is also considered. The result is obtained by asymptotic expansions of some transition functions. For this we use some new asymptotic expansion of the Bessel function.

Weighted quantile correlation tests for Gumbel, Weibull and Pareto families

Csörgõ S., Szabó T.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 2

227--250

Weighted quantile correlation tests are worked out for the Gumbel location and location-scale families. Our theoretical emphasis is on the determination of computable forms of the asymptotic distributions under the null hypotheses, which forms are based on the solution of an associated eigenvalue-eigenfunction problem. Suitable transformations then yield corresponding composite goodness-of-fit tests for the Weibull family with unknown shape and scale parameters and for the Pareto family with an unknown shape parameter. Simulations demonstrate slow convergence under the null hypotheses, and hence the inadequacy of the asymptotic critical points. Other rounds of extensive simulations illustrate the power of all three tests: Gumbel against the other extreme-value distributions, Weibull against gamma distributions, and Pareto against generalized Pareto distributions with logarithmic slow variation.

The speed of convergence of random products of sums of independent random variables

Krajka T., Rychlik Z.

Journal of Mathematics and Applications

2009

Vol. 31

51-66